Finite Group Databasebeta

← C7×#10248 #13187464

Subgroups — C7×#10248 #13187464

Order 1792.

604 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3209C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic
3C7×#2542C211yesmaximal, characteristic
4C7×#1548C211yesmaximal, characteristic, complemented
5C7×#1503C211yesmaximal, characteristic, complemented
6C7×#1478C211yesmaximal, characteristic
7C7×#1303C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C22⋊Q8—22no—
11C2×C4×C7×D4C2211yescharacteristic
12C7×#494—21no—
13C7×#487—21no—
14C7×#478—22no—
15C7×#476—21no—
16C7×#474—21no—
17C7×#462—22no—
18C7×#451—22no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#428—21no—
22C7×#423C2211yescharacteristic
23C7×#418—21no—
24C7×#399—22no—
25C7×#390—21no—
26C7×#384—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C7×#333—22no—
29C7×#328—22no—
30C7×D4⋊C8—22no—
31#10248C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D412yes—
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C2×C7×Q16—24no—
36C2×C7×M16—21no—
37C7×C4⋊Q8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D411yescharacteristic
43C7×#125—21no—
44C7×C22⋊Q8—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—48no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—24no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—24no—
60C2×C7×C22⋊C4—21no—
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63C7×#107—22no—
64C7×#106—22no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×C4⋊C8D412yes—
68C7×C4⋊C8D412yes—
69C7×#103—24no—
70C7×#102—22no—
71C7×#102—22no—
72C7×C22⋊C8—21no—
73C7×C22⋊C8—21no—
74C4×C7×C8—21no—
75C4×C7×C8—21no—
76C7×#95D412yes—
77#3209C2×C711yescharacteristic, complemented
78#3077C2×C711yescharacteristic
79#2542C2×C711yescharacteristic
80#1548C2×C711yescharacteristic, complemented
81#1503C2×C711yescharacteristic, complemented
82#1478C2×C711yescharacteristic
83#1303C2×C711yescharacteristic, complemented
84C24×C7—21no—
85C2×C7×Q8—22no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—22no—
89C2×C7×Q8—44no—
90C2×C7×D4—21no—
91C2×C7×D4—21no—
92C2×C7×D4—21no—
93C2×C7×D4—21no—
94C2×C7×D4—21no—
95C2×C7×D4Q1612yescomplemented
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—21no—
99C22×C4×C7—44no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C7×Q16—84no—
106C7×M16—41no—
107C7×M16—41no—
108C2×C7×C8—22no—
109C2×C7×C8—22no—
110C2×C7×C8—41no—
111C7×C4⋊C4—24no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—48no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C22⋊C4—48no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C42×C7—41no—
130C42×C7—41no—
131C42×C7—41no—
132C42×C7C2×D411yescharacteristic
133C42×C7C2×D411yescharacteristic
134C42×C7C2×D411yescharacteristic
135C42×C7#4111yescharacteristic
136#533C22×C711yescharacteristic
137C2×C4⋊Q8C22×C711yescharacteristic
138C2×C22⋊Q8—22no—
139C2×C4×D4C22×C711yescharacteristic
140#494—21no—
141#487—21no—
142#478—22no—
143#476—21no—
144#474—21no—
145#462—22no—
146#451—22no—
147#431C22×C711yescharacteristic
148#431C22×C711yescharacteristic
149#428—21no—
150#423C22×C711yescharacteristic
151#418—21no—
152#399—22no—
153#390—21no—
154#384—21no—
155C4×C4⋊C4C22×C711yescharacteristic
156#333—22no—
157#328—22no—
158D4⋊C8—22no—
159C23×C7—22no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7#12011yescharacteristic
165C7×Q8—42no—
166C7×Q8—44no—
167C7×Q8—81no—
168C7×Q8—82no—
169C7×D4—22no—
170C7×D4—22no—
171C7×D4—41no—
172C7×D4—41no—
173C2×C4×C7—22no—
174C2×C4×C7—24no—
175C2×C4×C7—41no—
176C2×C4×C7—42no—
177C2×C4×C7—42no—
178C2×C4×C7—44no—
179C2×C4×C7—42no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—42no—
184C2×C4×C7—82no—
185C2×C4×C7—82no—
186C2×C4×C7#12011yescharacteristic
187C2×C4×C7#12011yescharacteristic
188C2×C4×C7#12111yescharacteristic
189C2×C4×C7#12111yescharacteristic
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12711yescharacteristic
192C2×C4×C7C2×Q1611yescharacteristic
193C2×C4×C7#13511yescharacteristic
194C2×C4×C7#13711yescharacteristic
195C2×C4×C7#13711yescharacteristic
196C7×C8—42no—
197C7×C8—42no—
198C7×C8—81no—
199C22×Q8C7×D412yes—
200C22×D4C7×D411yescharacteristic
201C23×C4—21no—
202C2×Q16—24no—
203C2×M16—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C7×D411yescharacteristic
209C4⋊Q8C7×D411yescharacteristic
210#125—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—48no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220#117—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C22⋊C4—24no—
227C2×C22⋊C4—21no—
228C2×C42—21no—
229C2×C42C23×C711yescharacteristic
230#107—22no—
231#106—22no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8C7×D412yes—
235C4⋊C8C7×D412yes—
236#103—24no—
237#102—22no—
238#102—22no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243#95C7×D412yes—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—44no—
247C22×C7—42no—
248C22×C7—82no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7#39211yescharacteristic
252C22×C7#45111yescharacteristic
253C22×C7#45211yescharacteristic
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—21no—
257C4×C7—44no—
258C4×C7—81no—
259C4×C7—82no—
260C4×C7—81no—
261C4×C7—81no—
262C4×C7#45111yescharacteristic
263C4×C7#45211yescharacteristic
264C4×C7#45312yes—
265C4×C7#46711yescharacteristic
266C4×C7#47011yescharacteristic
267C24—21no—
268C2×Q8—22no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—44no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4C7×Q1612yescomplemented
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4C2×C7×D411yescharacteristic
286C22×C4C2×C7×D411yescharacteristic
287C22×C4C2×C7×D411yescharacteristic
288Q16—84no—
289M16—41no—
290M16—41no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—41no—
294C4⋊C4—24no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—48no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C22⋊C4—48no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42C2×C7×D411yescharacteristic
316C42C2×C7×D411yescharacteristic
317C42C2×C7×D411yescharacteristic
318C42C7×#4111yescharacteristic
319C2×C7—21no—
320C2×C7—21no—
321C2×C7—82no—
322C2×C7#200111yescharacteristic
323C2×C7#200211yescharacteristic
324C2×C7#200911yescharacteristic
325C23—22no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23C7×#12011yescharacteristic
331Q8—42no—
332Q8—44no—
333Q8—81no—
334Q8—82no—
335D4—22no—
336D4—22no—
337D4—41no—
338D4—41no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4C7×#12011yescharacteristic
353C2×C4C7×#12011yescharacteristic
354C2×C4C7×#12111yescharacteristic
355C2×C4C7×#12111yescharacteristic
356C2×C4C7×#12111yescharacteristic
357C2×C4C7×#12711yescharacteristic
358C2×C4C2×C7×Q1611yescharacteristic
359C2×C4C7×#13511yescharacteristic
360C2×C4C7×#13711yescharacteristic
361C2×C4C7×#13711yescharacteristic
362C8—42no—
363C8—42no—
364C8—81no—
365C7#1024811yescharacteristic, direct product factor, Sylow
366C22—21no—
367C22—21no—
368C22—44no—
369C22—42no—
370C22—82no—
371C22—81no—
372C22—81no—
373C22C7×#39211yescharacteristic
374C22C7×#45111yescharacteristic
375C22C7×#45211yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—44no—
380C4—81no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4C7×#45111yescharacteristic
385C4C7×#45211yescharacteristic
386C4C7×#45312yes—
387C4C7×#46711yescharacteristic
388C4C7×#47011yescharacteristic
389C2—21no—
390C2—21no—
391C2—82no—
392C2C7×#200111yescharacteristic
393C2C7×#200211yescharacteristic
394C2C7×#200911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C7×#3209C2semidirect
4. C7×#1548C2semidirect
5. C7×#1503C2semidirect
7. C7×#1303C2semidirect
31. #10248C7direct (N × H)
77. #3209C2×C7semidirect
80. #1548C2×C7semidirect
81. #1503C2×C7semidirect
83. #1303C2×C7semidirect
95. C2×C7×D4Q16semidirect
278. C2×D4C7×Q16semidirect
365. C7#10248direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C7×#32092
C7×#30772
C7×#25422
C7×#15482
C7×#15032
C7×#14782
C7×#13032
#102487